This work investigates two types of inverse interface problems in scenarios where only very limited Cauchy data is available. These problems are associated with the Laplace equation featuring a Robin-type flux jump across an internal interface. The first problem focuses on reconstructing the location of cracks along a known interface using Cauchy data measured on the outer boundary. The second problem involves determining the location of an unknown interface based on Cauchy data from the outer boundary. To address these challenges, we adopt an efficient Direct Sampling Method (DSM) and introduce innovative enhancements to the boundary conditions in the reference system, thereby maximizing the utility of the available Cauchy data. Additionally, we propose a novel refinement to further improve the robustness of the DSM against noise. We provide a detailed exposition of the general principles underlying the DSM and systematically present its computational implementation steps. Through detailed Fourier analysis and computations, we illustrate the theoretical background of the DSM as well as the effectiveness of our refinement approach. A series of numerical experiments demonstrates that our method yields highly satisfactory results, even when processing incomplete and noisy Cauchy data on the outer boundary. We introduce quantitative metrics, such as Mean Localization Error (MLE) and Contrast-to-Noise Ratio (CNR), to rigorously evaluate the performance of our method. These findings underscore the exceptional effectiveness and broad applicability of the proposed approach.

We prove the uniqueness in determining a spatially varying zeroth-order coefficient of a one-dimensional time-fractional diffusion equation by initial value and Cauchy data at one end point of the spatial interval.

In this paper we discuss the Anderson's type acceleration method for numerical optimizations. Most mathematical modeling problems can be formulated as constrained optimization. The necessary optimality condition is written as a fixed point problem in a Banach space. Anderson's acceleration method improves the convergence of the standard fixed point iteration by minimizing the total sum of residuals and updating solutions through an optimal linear combination of a sequence of iterates. Thus, it is a form of iterative method of retard which uses the history of solutions as a reduced order element method (ROM). The weights are determined optimally by the least squares problem based on the total residual. We analyze Anderson's method and the reduced order method (ROM) for nonlinear least squares problems of minimizing |F(x)| squared. That is, the solution is approximated by a linear combination of sequentially generated solutions, and then we minimize the equation error on the linear manifold spanned by the iterates. We use the reduced order Gauss Newton method to solve the least squares problem for |F(x)| squared on the linear solution manifold. For the linear equation case it is similar to Anderson's method. Anderson's method approximates the solution to the nonlinear ROM. We consider variable step size gradient and quasi Newton methods and the variable fixed point iteration to generate the solution basis, then combine these with ROM acceleration. It converges very rapidly if the condition number of matrix A is not large. The variable iterate with ROM acceleration is nearly optimal, and ROM also regularizes and stabilizes the convergence. We also consider a randomly overlapped Kaczmarz type method and develop an acceleration approach for large scale ill posed linear systems. Finally, we analyze the convergence of the ROM to the operator equation.

In this work, we propose an innovative iterative direct sampling method to solve nonlinear elliptic inverse problems from a limited number of pairs of Cauchy data. It extends the original direct sampling method (DSM) by incorporating an iterative mechanism, enhancing its performance with a modest increase in computational effort but a clear improvement in its stability against data noise. The method is formulated in an abstract framework of operator equations and is applicable to a broad range of elliptic inverse problems. Numerical results on electrical impedance tomography, optical tomography and cardiac electrophysiology etc. demonstrate its effectiveness and robustness, especially with an improved accuracy for identifying the locations and geometric shapes of inhomogeneities in the presence of large noise, when compared with the standard DSM.

Inverse medium scattering problems arise in many applications, but in practice, the measurement data are often restricted to a limited aperture by physical or experimental constraints. Classical sampling methods, such as MUSIC and the linear sampling method, are well understood for full-aperture data, yet their performance deteriorates severely under limited-aperture conditions, especially in the presence of noise. We propose a new sampling method tailored to the inverse medium problem with limited-aperture data. The method is motivated by the linear sampling framework and incorporates a weight function into the index function. The weight is designed so that the modified kernel reproduces the full-aperture behavior using only limited data, which both localizes oscillations and improves the conditioning of the far-field system, thereby yielding more accurate and stable reconstructions. We provide a theoretical justification of the method under the Born approximation and an efficient algorithm for computing the weight. Numerical experiments in two and three dimensions demonstrate that the proposed method achieves greater accuracy and robustness than existing sampling-type methods, particularly for noisy, limited-aperture data.

In this paper we discuss inverse medium problems. We develop the direct sampling method based on probing indices using the saddle point formulation. The medium is constructed by solutions of saddle point problems. The method improves the probing functions for the direct sampling method and directly images the medium. The method is very efficient and can be applied to a general class of inverse medium problems.

In this paper, we propose a direct probing method for the inverse problem based on the Eikonal equation. For the Eikonal equation with a point source, the viscosity solution represents the least travel time of wave fields from the source to the point at the high-frequency limit. The corresponding inverse problem is to determine the inhomogeneous wave-speed distribution from the first-arrival time data at the measurement surfaces corresponding to distributed point sources, which is called transmission travel-time tomography. At the low-frequency regime, the reconstruction approximates the frequency-depend wave-speed distribution. We analyze the Eikonal inverse problem and show that it is highly ill-posed. Then we developed a direct probing method that incorporates the solution analysis of the Eikonal equation and several aspects of the velocity models. When the wave-speed distribution has a small variation from the homogeneous medium, we reconstruct the inhomogeneous media using the filtered back projection method. For the high-contrast media, we assume a background medium and develop the adjoint-based back projection method to identify the variations of the medium from the assumed background.

In this paper, we study the parabolic Dirichlet boundary optimal control on complex connected domains. It is well known that both the complex connected domain and the Dirichlet boundary control bring great difficulties to theoretical analysis and numerical calculation. The complex connected domain is a typical non-convex domain, and it is difficult for the traditional numerical method to obtain the same convergence order as the state and adjoint state for the Dirichlet boundary control. The optimal system of the proposed control problem is first obtained by using Lagrange multiplier method. Then, based on the variational form, the Fourier finite volume element method is used to obtain the fully discrete scheme of the optimality system, that is, using Fourier expansion method in azimuthal direction and applying finite volume element method in radial direction respectively, and the Crank-Nicolson scheme in the time direction. Next, we strictly prove the error estimates of state, adjoint state and Dirichlet boundary control by using the variational discretization concept. Finally, the theoretical analysis results and the feasibility and effectiveness of the proposed method are verified by numerical experiments.

We study an optimal control problem governed by elliptic PDEs with interface, which the control acts on the interface. Due to the jump of the coefficient across the interface and the control acting on the interface, the regularity of solution of the control problem is limited on the whole domain, but smoother on subdomains. The control function with pointwise inequality constraints is served as the flux jump condition which we called Neumann interface control. We use a simple uniform mesh that is independent of the interface. The standard linear finite element method can not achieve optimal convergence when the uniform mesh is used. Therefore the state and adjoint state equations are discretized by piecewise linear immersed finite element method (IFEM). While the accuracy of the piecewise constant approximation of the optimal control on the interface is improved by a postprocessing step which possesses superconvergence properties; as well as the variational discretization concept for the optimal control is used to improve the error estimates. Optimal error estimates for the control, suboptimal error estimates for state and adjoint state are derived. Numerical examples with and without constraints are provided to illustrate the effectiveness of the proposed scheme and correctness of the theoretical analysis.

We present a two-stage least-squares method to inverse medium problems of reconstructing multiple unknown coefficients simultaneously from noisy data. A direct sampling method is applied to detect the location of the inhomogeneity in the first stage, while a total least-squares method with mixed regularization is used to recover the medium profile in the second stage. The total least-squares method is designed to minimize the residual of the model equation and the data fitting, along with an appropriate regularization, in an attempt to significantly improve the accuracy of the approximation obtained from the first stage. We shall also present an analysis on the well-posedness and convergence of this algorithm. Numerical experiments are carried out to verify the accuracies and robustness of this novel two-stage least-squares algorithm, with great tolerance of noise.

In this paper we propose a least squares formulation for ill-posed inverse problems. For example, ill-posed inverse problems in partial differential equations are those such that a solution of inverse problem exists for a smooth data but it does not depend continuously on data, or there is no solution for inverse problem, e.g. the Cauchy problem for elliptic equations and backward solution of parabolic equations. We develop the least squares formulation in which the sum of equations error over domain and data fitting criterion and Tikhonov regularization terms is minimized over entire solutions. In this way we can establish the existence and uniqueness of an inverse solution and establish the continuity of the inverse solution for noisy data in L-2. The method can be applied to a general class of non-linear inverse problems and an operator theoretic stability analysis is developed. We describe how one can apply the method for various PDE inverse problems. Numerical tests using backward heat equation and Cauchy problem for elliptic equations are presented to demonstrate the applicability and performance of the proposed method.

We present a two-stage least-squares method for inverse medium problems of reconstructing multiple unknown coefficients simultaneously from noisy data. A direct sampling method is applied to detect the location of the inhomogeneity in the first stage, while a total least-squares method with a mixed regularization is used to recover the medium profile in the second stage. The total least-squares method is designed to minimize the residual of the model equation and the data fitting, along with an appropriate regularization, in an attempt to significantly improve the accuracy of the approximation obtained from the first stage. We shall also present an analysis on the well-posedness and convergence of this algorithm. Numerical experiments are carried out to verify the accuracies and robustness of this novel two-stage least-squares algorithm, with high tolerance of noise in the data.

An accurate semi-decoupling numerical method has been proposed for nonlinear singular differential equation with interface, which combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to decouple the singular interface problem and get high order accurate numerical solution. Key to our proposed new method is introducing three augmented variables involving the interface and the singularity, and reconstructing the representative of the solution as the Puiseux series expansions on the interface to decouple original problem as two standard nonlinear singular problems with the second jump condition. In this way, the augmented variables related with semi-analytic solutions near the interface and numerical solutions can be simultaneously solved in the remaining interval on both sides of the interface. It demonstrates that our method does not take more heavier works for handling jump conditions like other methods, and is independent of the interface and jump ratio. A rigorous error estimate for the solution of nonlinear singular differential equation with interface and augmented variables is obtained. Numerical experiments for those singular differential equations with interface confirm the theoretical analysis and accuracy of the new approach. In particular, an interesting example with blow-up coefficient at singular point shows that our approach can be extended to numerically solve strongly singular interface problem.

In this paperIto, K. the sensitivity analysis is discussed for the parameter-dependent optimization and constraint optimization. The sensitivity of the optimality value function with respect to the change in parameters plays a significant role in the inverse problems and the optimization theory, including economics, finance, the Hamilton–Jacobi theory, the inf-sup duality and the topological design and the bi-level optimization. We develop the calculus for the value function and present its applications in the variational calculus, the bi-level optimization and the optimal control and optimal design, shape calculus and inverse problems.

This paper introduces a computationally efficient data assimilation scheme based on Gaussian quadrature filtering that potentially outperforms current methods in data assimilation for moderately nonlinear systems. Moderately nonlinear systems, in this case, are systems with numerical models with small fourth and higher derivative terms. Gaussian quadrature filters are a family of filters that make simplifying Gaussian assumptions about filtering pdfs in order to numerically evaluate the integrals found in Bayesian data assimilation. These filters are differentiated by the varying quadrature rules to evaluate the arising integrals. The approach we present, denoted by Assumed Gaussian Reduced (AGR) filter, uses a reduced order version of the polynomial quadrature first proposed in Ito and Xiong [2000. Gaussian filters for nonlinear filtering problems. IEEE Trans. Automat. Control. 45, 910-927]. This quadrature uses the properties of Gaussian distributions to form an effectively higher order method increasing its efficiency. To construct the AGR filter, this quadrature is used to form a reduced order square-root filter, which will reduce computational costs and improve numerical robustness. For cases of sufficiently small fourth derivatives of the nonlinear model, we demonstrate that the AGR filter outperforms ensemble Kalman filters (EnKFs) for a Korteweg-de Vries model and a Boussinesq model.

In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the H 2 -norm and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to oblique derivative boundary conditions. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

We measured the Kr photoelectron spectrum in the region close to the 3p ionization threshold. Our high-resolution measurements allowed a clear observation of spectral structures due to electron correlation effects. Analysis based on relativistic multiconfiguration calculations could explain these observed peaks as due to strong configuration interactions between the 3p(-1) state and 3d(-2) nl states. Calculated and experimental data for peak assignments and intensity distributions are in good agreement. In addition, we measured the anisotropy parameter beta, which also agreed well with theory. These findings provide a detailed view of strong configuration interactions between the 3(-1) and 3d(-2)nl inner-shell hole states.

In this work, we develop efficient solvers for linear inverse problems based on randomized singular value decomposition (RSVD). This is achieved by combining RSVD with classical regularization methods, e.g., truncated singular value decomposition, Tikhonov regularization, and general Tikhonov regularization with a smoothness penalty. One distinct feature of the proposed approach is that it explicitly preserves the structure of the regularized solution in the sense that it always lies in the range of a certain adjoint operator. We provide error estimates between the approximation and the exact solution under canonical source condition, and interpret the approach in the lens of convex duality. Extensive numerical experiments are provided to illustrate the efficiency and accuracy of the approach.

We investigate spectral theory for a one-body Stark Hamiltonian under minimum regularity and decay conditions on the potential (actually allowing sub-linear growth at infinity). Our results include Rellich's theorem, the limiting absorption principle, radiation condition bounds and Sommerfeld's uniqueness, and most of these are stated and proved in sharp form employing Besov-type spaces. For the proofs we adopt a commutator scheme by Ito–Skibsted [13]. A feature of the paper is a special choice of an escape function related to parabolic coordinates, which conforms well with classical mechanics for the Stark Hamiltonian. The whole setting of the paper, such as the conjugate operator and the Besov-type spaces, is generated by this single escape function. We apply our results in the sequel paper [5].